Resolution of Singularities, Asymptotic Expansions of Integrals Over Sublevel Sets, and Applications
نویسنده
چکیده
Here φ(x) is a smooth bump function defined on a neighborhood of the origin and λ is a real parameter whose absolute value we assume to be large. If ∇f(0) 6= 0, then by repeated integrations by parts (see [S] Ch 8 for example), for any N one has an estimate |Iλ| < Cf,φ,N |λ|−N for appropriate constants Cf,φ,N . In the case where ∇f(0) = 0, that is, when f has a critical point at the origin, it can be proven (see [M]) using Hironaka’s resolution of singularities [H1]-[H2] that if the support of φ is contained in a sufficiently small neighborhood of the origin, then as λ→∞, Iλ has an asymptotic expansion of the form
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